The Solution of the Pyramid Problem Part 9

22 37 1151 13 12 5 5 1 11 18 3575

25 3 2727 85 77 36 9 2 12 31 4363

25 59 2122 89 80 39 13 3 12 59 4061

28 4 2094 17 15 8 4 1 14 2 1047

30 30 3649 65 56 33 11 3 15 15 1824

31 53 2685 53 45 28 7 2 15 56 4342

36 52 1165 5 4 3 3 1 18 26 582

41 6 4352 73 55 48 8 3 20 33 2176

42 4 3008 97 72 65 13 5 21 2 1504

43 36 1015 29 21 20 5 2 21 48 507

46 23 4985 29 20 21 7 3 23 11 5492

47 55 2992 97 65 72 9 4 23 57 4496

48 53 1648 73 48 55 11 5 24 26 3824

53 7 4835 5 3 4 2 1 26 33 5417

58 6 3315 53 28 45 9 5 29 3 1657

59 29 2351 65 33 56 7 4 29 44 4175

61 55 3906 17 8 15 5 3 30 57 4953

64 0 3878 89 39 80 8 5 32 0 1939

64 56 3273 85 36 77 11 7 32 28 1636

67 22 4849 13 5 12 3 2 33 41 2424

71 4 3129 37 12 35 7 5 35 32 1564

73 44 2327 25 7 24 4 3 36 52 1163

75 45 090 65 16 63 9 7 37 52 3045

77 19 1063 41 9 40 5 4 38 39 3531

79 36 4011 61 11 60 6 5 39 48 2005

81 12 931 85 13 84 7 6 40 36 465

83 16 138 145 17 144 9 8 41 38 069

87 12 2030 841 41 840 21 20 43 36 1015

Reference to the plan ratio table at the commencement, and to the tables here introduced, will shew that most of the primary triangles mentioned are indicated on the plan ratio table princ.i.p.ally by the lines corresponding to the ratios of the satellites. Thus--

PRIMARY TRIANGLE INDICATED BY

17, 144, 145. Triangle FP, PA, AF on plan.

13, 84, 85. Plan ratio of SJ to SU, 7 to 6.

11, 60, 61. Plan ratio BC to FB, 6 to 5, and DN to NR, 61 to 60.

12, 35, 37. Plan ratio EO to AY, 37 to 12, and EA to AY, 35 to 12.

5, 12, 13. Plan ratio CY to BC, 3 to 2; JE to EX, 3 to 2; CA to YA, 5 to 1; and NZ to ZA, 12 to 5.

8, 15, 17. Plan ratio FB to BY, 5 to 3, and AC to BC, 15 to 8.

33, 56, 55. Plan ratio YX to AY, 7 to 4; AB to BO, 7 to 4; and EA to AZ, 7 to 4.

28, 45, 53. Exists on plan, AB, BJ, JA.

3, 4, 5. Pervades the plan, and is also indicated by plan ratio GX to DG, 2 to 1; SU to SV, 2 to 1; and CY to YZ, 3 to 1.

48, 55, 73. Exists on plan, FW, WV, VF--and is also indicated by plan ratio FO to OZ, 8 to 3.

65, 72, 97. Plan ratio AC to CH, 9 to 4; MY to YZ, 9 to 4.

20, 21, 29. Exists on plan FB, BA, AF; and plan ratio, GU to DG, 5 to 2.

It seems probable that could I add to my pyramid plan the lines and triangles that the missing eleven pyramids would supply, it would comprise a complete table on which would appear indications of all the ratios and triangles made use of in right-angled trigonometry, a "_ratiometer_" in fact.

I firmly believe that so far as I have gone it is correct--and it is possible, therefore, with the start that I have made, for others to continue the work, and add the eleven pyramids to the plan in their correct geometrical position. By continuing the system of evolution by which I defined the position of Cephren, and the little pyramid to the south-east of Cheops, after I had obtained Cheops and Mycerinus, may be rebuilt, at one and the same time, a skeleton of the trigonometrical tables of a forgotten civilization, and the plan of those pyramids which are its only link with the present age.

-- 13. THE SIZE AND SHAPE OF THE PYRAMIDS INDICATED BY THE PLAN.

I pursued my investigations into the slopes and alt.i.tudes of the pyramids without reference to the plan, after once deciding their exact bases.

Now it will be interesting to note some of the ways in which the plan hints at the shape and size of these pyramids, and corroborates my work.

The dimensions of _Cheops_ are indicated on the plan by the lines EA to YA, measuring 840 and 288 R.B. cubits respectively, being the half periphery of its horizontal section at the level of Cephren's base, and its own alt.i.tude from its own base. (_See Fig_. 5.)

The line EA, in fact, represents in R.B. cubits the half periphery of the bases of either Cheops or Cephren measured at the level which I have set forth as the _plan level_, viz., base of Cephren.

The ratio of Cephren's base to Cephren's alt.i.tude is indicated on the plan by the ratios of the lines BC to EB, or FO to OR, viz., 32 to 21.

(_See Fig._ 4.)

The alt.i.tude of Mycerinus above Cephren's base appears on plan in the line EF, measuring 136 R.B. cubits.

The line EO on plan measures 888 cubits, which would be the length of a line stretched from the apex of Cheops to the point E, at the level of Cheops' base.

This merits consideration:--the lines EA and AY are connected on plan at the centre of Cheops, and the lines EO and EA are connected on plan at the point E.

Now the lines EO, EA and AY are sides of a "primary triangle," whose ratio is 37, 35, 12, and whose measure in cubits is 888, 840, and 288; and if we suppose the line EA to be stretched horizontally beneath the pyramids at the level of the base of Cheops from E to A on plan, and the line AY to be a plumb line hanging from the apex of Cheops to the level of his base, then will the line EO just stretch from the point E to the apex of Cheops, and the three lines will connect the two main pyramids by a vertical triangle of which EA, AY and EO form the base, perpendicular, and hypotenuse. Or, to explain it in another manner: let the line EA be a _cord_ stretching horizontally from A at the centre of the base of Cheops to the point E, both ends being at the same level; let the line AY be a _rod_, lift it on the end A till it stands erect, then is the end Y the apex of Cheops. Now, the line EO would just stretch from the top of the rod AY to the point E first described.

It is a singular coincidence, and one that may be interesting to students of the _interior_ of the Pyramids, that the side EP, of the small 3, 4, 5 triangle, EP, PF, FE, in the centre of the plan, measures 8160 R.B. cubits, which is very nearly eight times the "_true breadth_"